Thursday, January 16, 2020

Locus of Points

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(C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

There's a step ladder in the garage ... pity that it isn't closer.




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Tuesday, January 14, 2020

Unit Rates

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(C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

If you know what I Mina. Or, mean-a?




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Friday, January 10, 2020

Blog: Zeroth Power and Fractional Exponents

I'm starting a new, old math book (report to come), and I came across something interesting in the first chapter about numbers: another way to look at why N0 = 1.

The rules for exponents are simple:

Na * Nb = N(a + b)
and Na / Nb = N(a - b)

In the case of multiplying repeatedly by the same number:

Na * N = N(a + 1)

So, say, 24 * 2 = 24 + 1 = 25
and 25 * 2 = 25 + 1 = 26

Likewise, when dividing, repeatedly, by the same factor, the exponent would be decremented:

Thus, 23 / 2 = 23 - 1 = 22
and 22 / 2 = 22 - 1 = 21
and, finally, 21 / 2 = 21 - 1 = 20.

So what is 20?
If we evaluate the other expressions, we have the following:
8 / 2 = 4; 4 / 2 = 2; 2 / 2 = 1. So 20 = 1.

Note that we could continue the progression into negative exponents if we keep dividing. This will lead to fractions. Perhaps on another day...

Fractional exponents

What does it mean to have a fraction as an exponent?

Keep in mind that negative exponents have nothing to do with negative numbers. They create fractions. So fractional exponents won't create fractions.

Let's review one more rule about exponents:


(Na)b = N(ab)

So (23)4 = 2(3*4) = 212 = 4096.

What if there is an exponent of 2/3 or 3/5? First, consider that 2/3 = (2)(1/3) and 3/5 = (3)(1/5). The 2 and 3 still mean the second and third powers, but what about the unit fractions of 1/3 or 1/5?

If I take the positive square root, which I'll abbreviate SQRT(), of N2, I will get N, because N * N = N2.
If I take the positive square root of N4, I will get N2, because N2 * N2 = N4, etc.

So SQRT(Na) will give (N(1/2)a).
However, our rule tells us that (N(1/2)a) = (Na)1/2.
So (Na)1/2 is another way to write SQRT(Na), and N1/2 is another way to write SQRT(N).

Fractions and Zero

Consider the following progression:

SQRT(16) = 4
SQRT( SQRT(16)) = 2
SQRT( SQRT( SQRT(16))) = 1.414...
SQRT( SQRT( SQRT( SQRT(16)))) = 1.189...
SQRT( SQRT( SQRT( SQRT( SQRT(16))))) = 1.090...

As you repeatedly take the square root, the answer will get closer and closer to 1.
If we rewrite that using exponents, we get the following:

16(1/2) = 4
(16(1/2))(1/2) = 16(1/4) = 2
((16(1/2))(1/2))(1/2) = 16(1/8) = 1.414...
(((16(1/2))(1/2))(1/2))(1/2) = 16(1/16) = 1.189...
((((16(1/2))(1/2))(1/2))(1/2))(1/2) = 16(1/32) = 1.090...

As the denominator gets larger, the fraction gets smaller. As the denominator goes toward infinity, the fraction goes toward zero. And the value on the right side of the equal sign goes toward 1.

Wednesday, January 08, 2020

Non-abelian Group

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(C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

For years, I have resisted making abelian group jokes because I don't want to explain them. I still don't.

Trust me, there are a handful of people laughing hysterically (or chuckling politely) at this, and I hope that they will share it with others who do not require explanation.

However, for those who would like more information, allow me to point to this tweet from the account Great Women of Mathematics.




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Monday, January 06, 2020

I vs i

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(C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Know what I talkin' bout?

A long time ago, I did a "Pop i" that had big sailor forearms, but then the Roman I would've needed arms, too, but not a Gladius

I did a Pi vs. Pi comic years ago. I never did draw the followup.




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Thursday, January 02, 2020

End of Year / End of Decade Report

I wanted to do this before the end of the year, but I was a little busy, and that was a good thing. I ended the year -- and decade -- on a happy note.

This made me happy because for the past month I've read others' account of the achievements and milestones of the past ten years. And that's where I ran into problems.

Thinking of achievements was difficult when they were weighed down by loss. I'm at an age when most of my achievements are behind me while the next generation steps out and marks their own milestones. On the other end of the spectrum, the older generation ... well, we know what eventually happens when you get older. You, one day, stop getting older. There was a lot of that in the past decade, and -- possibly a statistical anomaly, or just the edge of normal -- many of them fell within an eight-month period. That wasn't a great year.

Professionally, I was excessed from a position I held for more that ten years. Then I made it back, and I was let go again. I've bounced around the system since, including what I thought was a new permanent position, only to be let go, again, at the end of the term. Nothing that I did -- they loved me -- but they all play budget games. They love me when they're desperate, but when they catch their breath -- and I've done the hard part -- they'll look for someone with under two years experience who they can push around more and pay less.

But, yes, there were some positive things that I can focus on:

For starters, I'm still writing this blog, and still creating comics. These both started the decade before last, but the fact that they continue shows a great commitment (even if Wikipedia still won't list my twelve-year-old webcomic).

For another, nine of those comics made it into a Logic textbook at the University of Sweden. It's not the "sexiest" thing I've been published in, but I'd like to think college students are getting a chuckle (or a groan) from some of Mr. Keegan's exploits.

Also professionally related, there was a book proposal called Fueled By Coffee & Love that was to be filled with teacher stories. I honestly didn't know what that meant, and the guidelines were so open, I wasn't sure what, or how much, to write. After the first book came out and a sequel was proposed, I was probably one of the first to submit, and "My Teaching Journey" was accepted for publication. (Side note: it reminded me how much I had to owe Tracy S. for me still being a teacher.)

But fiction writing? That has been on the back burner for the past 20-plus years. Many excuses have been made. And the few times I sat down to type and finished something, nothing came of it. That changed with a chance meeting with Danielle Ackley-McPhail, a writer, editor, and owner or eSpec Books. We're just friends, acquaintances, really, who see each other a couple times per year at sci-fi conventions.

I started reading the company blog, and I saw their first flash fiction contest. I submitted something under a thousand words long (it could have been up to twice as long) and it won! Well, it co-won. Danielle told me she liked my story because it had the most "catty-ness" of all the entries (and that was the theme).

After that, I tried to enter as often as I could, usually writing in the final days of the month. Sadly, I discovered that a story that she liked was the only entry that month. It was a little deflating, but the positive feedback helped. One story, "Cyber Where?", Danielle called the best story I'd written. I was elated to hear that. It meant I was improving, growing. Going somewhere.

This brings me to the past year or so. I tried writing flash fiction for a handful of markets out there. Not of them would pay more that twenty dollars, if that much, for 1,000 words or less. So far, I haven't been able to crack any of them.

I kept at them because I couldn't focus enough to write a short story, putting 3,000 to 5,000 words together. I was thinking in terms of scenes, and not acts.

By the time December rolled around, I had a plan of attack. A few months ago, there was a Kickstarter by Zombies Need Brains for three new anthologies. I back it for all three ebooks. Part of the plan was to have slots available for submissions. The odds aren't great, given the size of the slush pile, but a good, well-written story makes a difference.

I had ideas for two of the three anthologies. It came down to the final days, but I finished the two short stories I'd planned on, and both came in over 4,000 words. I wasn't sure that the second one would be written, but both had bounced around in my head for over a month, so I knew the acts as well as the scenes, and it wasn't just a bunch of dialogue.

I won't know until next month if either is accepted or both are rejected, but I'm happy that I got through it.

And if that wasn't enough of a high note to end the year on, there is one final note, but I can't mention it. Not that it's hush-hush or anything (then again, maybe it is), but until I see something announced, I don't know for sure that it's definite. At least not right now. But good things are coming, if I can keep at them.

Wednesday, January 01, 2020

Happy New Year 2020

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(C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Happy New Year! May all your characters stay viable!

The funny thing about today's New Years Day comic is that it isn't the one I might have done a week ago. What would I have done? I have no idea. I forgot -- didn't write it down. But something came to me that I liked.

I've done binary jokes, along with other bases, and even a modulo (remainder) function. And factoring when it was interesting. (In this case, 101 wasn't so interesting, at least not as a new character.)

Every year, in the last week of December, I see a post or tweet from some math person I follow online with a list of "fun facts" about the number of the New Year. The "fact" is that there are so many of them, you can always find something.

For example: Let's say you wanted to find a bunch of consecutive numbers that add up to 2020. Then see if the answer to any of the following are whole numbers:

a + a + 1 = 2a + 1 = 2020
b + b + 1 + b + 2 = 3b + 3 = 2020
c + c + 1 + c + 2 + c + 3 = 4c + 6 = 2020
d + d + 1 + d + 2 + d + 3 + d + 4 = 5d + 10 = 2020
etc.

I had unnecessary notation when words are fine. You can see the progression. The coefficient increases by 1 and the constant is the next triangle number.

A quick check online yields the following:

402 + 403 + 404 + 405 + 406 = 2020
249 + 250 + 251 + 252 + 253 + 254 + 255 + 256 = 2020
etc.

Additionally, many numbers can be written as the the sum of two squares. Most can be written as the sum of three squares, and all can be written as a sum or difference of three squares. Moreover, every square is the sum of two triangular numbers, so that just expands the possibilities.

It looks great, but makes for stale comics. The formula I used today is incredibly arbitrary and created backward from the solution. It doesn't have any particular meaning.

As for the 20/20 vision jokes, along with the Barbara Walters gag, have been old for months now. Which is why I did one on Monday, and not today. Not going to toss it out just because it's old if I can find a way to use it.

In any case, thank you for being one of the blog readers. I appreciate the ones who take the time to come here and read the posts, instead of just looking at the comic on line. Even moreso the people who comment here or on social media. (Note: the social media comments may drive some traffic here but those comments will be lost in the bit-storm like tears in rain.)

Have a Happy New Year. Here's to hoping that there is at least 100 new comics before it's over.




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